Application of (Max, +)-Algebra to the Optimal Buffer Size in Poisson Driven Deterministic Queues in Series with Blocking

Abstract

In this study, by applying (max, +)-algebra to a stochastic event graph, a special case of timed Petri nets, we consider characteristics of waiting times in Poisson driven single-server 2 queues in series with a finite buffer and having constant service times at each queue. We show that the sojourn time does not depend on the finite buffer capacity and also derive the explicit expressions of waiting times at all areas of the system as a function of the finite buffer capacity, which allow one to compute and compare waiting times under two blocking policies. Moreover, an optimization problem which determines the smallest buffer capacity satisfying a predetermined probabilistic constraint on waiting times is considered as an application of these results.